## A Course on Borel SetsA Course on Borel sets provides a thorough introduction to Borel sets and measurable selections and acts as a stepping stone to descriptive set theory by presenting important techniques such as universal sets, prewellordering, scales, etc. It is well suited for graduate students exploring areas of mathematics for their research and for mathematicians requiring Borel sets and measurable selections in their work. It contains significant applications to other branches of mathematics and can serve as a self- contained reference accessible by mathematicians in many different disciplines. It is written in an easily understandable style and employs only naive set theory, general topology, analysis, and algebra. A large number of interesting exercises are given throughout the text. |

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### Contents

Cardinal and Ordinal Numbers | 1 |

12 Order of Inﬁnity | 4 |

13 The Axiom of Choice | 7 |

14 More on Equinumerosity | 11 |

15 Arithmetic of Cardinal Numbers | 13 |

16 WellOrdered Sets | 15 |

17 Transﬁnite Induction | 18 |

18 Ordinal Numbers | 21 |

44 The First Separation Theorem | 147 |

45 OnetoOne Borel Functions | 150 |

46 The Generalized First Separation Theorem | 155 |

47 Borel Sets with Compact Sections | 157 |

48 Polish Groups | 160 |

49 Reduction Theorems | 164 |

410 Choquet Capacitability Theorem | 172 |

411 The Second Separation Theorem | 175 |

19 Alephs | 24 |

110 Trees | 26 |

111 Induction on Trees | 29 |

112 The Souslin Operation | 31 |

113 Idempotence of the Souslin Operation | 34 |

Topological Preliminaries | 39 |

22 Polish Spaces | 52 |

23 Compact Metric Spaces | 57 |

24 More Examples | 63 |

25 The Baire Category Theorem | 69 |

26 Transfer Theorems | 74 |

Standard Borel Spaces | 80 |

32 BorelGenerated Topologies | 91 |

33 The Borel Isomorphism Theorem | 94 |

34 Measures | 100 |

35 Category | 107 |

36 Borel Pointclasses | 115 |

Analytic and Coanalytic Sets | 127 |

42 𝚺11 and 𝚷11 Complete Sets | 135 |

43 Regularity Properties | 141 |

412 CountabletoOne Borel Functions | 178 |

Selection and Uniformization Theorems | 183 |

51 Preliminaries | 184 |

52 Kuratowski and RyllNardzewskis Theorem | 189 |

53 Dubins Savage Selection Theorems | 194 |

54 Partitions into Closed Sets | 195 |

55 Von Neumanns Theorem | 198 |

56 A Selection Theorem for Group Actions | 200 |

57 Borel Sets with Small Sections | 204 |

58 Borel Sets with Large Sections | 206 |

59 Partitions into G𝜎 Sets | 212 |

510 Reflection Phenomenon | 216 |

511 Complementation in Borel Structures | 218 |

512 Borel Sets with σCompact Sections | 219 |

513 Topological Vaught Conjecture | 227 |

514 Uniformizing Coanalytic Sets | 236 |

241 | |

250 | |

253 | |

### Common terms and phrases

A C X A-measurable admits a Borel algebra analytic subsets assume Baire category theorem Baire property bijection Borel function Borel isomorphism Borel map Borel measurable Borel set Borel subset called Cantor cardinality Clearly closed set closed subset closed under countable coanalytic sets comeager completely metrizable continuous map contradiction convergent Corollary countable base countable intersections countable unions define deﬁned denote dense element equivalence classes Example Exercise f is continuous finite ﬁrst function f G N<N G NN Gg set Hence homeomorphic induction inﬁnite integer Lebesgue measurable Let f Let G map f meager measurable space nonempty open set nonmeager Note one-to-one open set ordinal numbers pairwise disjoint partition pointclass Polish group Polish space Polish topology Proposition prove real numbers result follows second countable selection theorem sequence Souslin operation Suppose Take topological space uncountable Polish space well-ordered set zero-dimensional

### Popular passages

Page 244 - Kechris and A. Louveau, Descriptive Set Theory and the Structure of Sets of Uniqueness, London Math. Soc. Lecture Note Ser., 128, Cambridge Univ.